11. THE CKM QUARK-MIXINGMATRIX11. CKM quark-mixing matrix 1 11. THE CKM QUARK-MIXINGMATRIX Revised...

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11. CKM quark-mixing matrix 1 11. THE CKM QUARK-MIXING MATRIX Revised February 2010 by A. Ceccucci (CERN), Z. Ligeti (LBNL), and Y. Sakai (KEK). 11.1. Introduction The masses and mixings of quarks have a common origin in the Standard Model (SM). They arise from the Yukawa interactions with the Higgs condensate, L Y = Y d ij Q I Li φd I Rj Y u ij Q I Li φ u I Rj +h.c., (11.1) where Y u,d are 3 × 3 complex matrices, φ is the Higgs field, i, j are generation labels, and is the 2 × 2 antisymmetric tensor. Q I L are left-handed quark doublets, and d I R and u I R are right-handed down- and up-type quark singlets, respectively, in the weak-eigenstate basis. When φ acquires a vacuum expectation value, φ = (0,v/ 2), Eq. (11.1) yields mass terms for the quarks. The physical states are obtained by diagonalizing Y u,d by four unitary matrices, V u,d L,R , as M f diag = V f L Y f V f R (v/ 2), f = u, d. As a result, the charged-current W ± interactions couple to the physical u Lj and d Lk quarks with couplings given by V CKM V u L V dL = V ud V us V ub V cd V cs V cb V td V ts V tb . (11.2) This Cabibbo-Kobayashi-Maskawa (CKM) matrix [1,2] is a 3 × 3 unitary matrix. It can be parameterized by three mixing angles and the CP -violating KM phase [2]. Of the many possible conventions, a standard choice has become [3] V = c 12 c 13 s 12 c 13 s 13 e s 12 c 23 c 12 s 23 s 13 e c 12 c 23 s 12 s 23 s 13 e s 23 c 13 s 12 s 23 c 12 c 23 s 13 e c 12 s 23 s 12 c 23 s 13 e c 23 c 13 , (11.3) where s ij = sin θ ij , c ij = cos θ ij , and δ is the phase responsible for all CP -violating phenomena in flavor-changing processes in the SM. The angles θ ij can be chosen to lie in the first quadrant, so s ij ,c ij 0. It is known experimentally that s 13 s 23 s 12 1, and it is convenient to exhibit this hierarchy using the Wolfenstein parameterization. We define [4–6] s 12 = λ = |V us | |V ud | 2 + |V us | 2 , s 23 = 2 = λ V cb V us , s 13 e = V ub = 3 (ρ + )= 3 ρ + i ¯ η) 1 A 2 λ 4 1 λ 2 [1 A 2 λ 4 ρ + i ¯ η )] . (11.4) These relations ensure that ¯ ρ + i ¯ η = (V ud V ub )/(V cd V cb ) is phase-convention-independent, and the CKM matrix written in terms of λ, A, ¯ ρ, and ¯ η is unitary to all orders in λ. The definitions of ¯ ρ, ¯ η reproduce all approximate results in the literature. For example, ¯ ρ = ρ(1 λ 2 /2+ ...) and we can write V CKM to O(λ 4 ) either in terms of ¯ ρ, ¯ η or, traditionally, V = 1 λ 2 /2 λ 3 (ρ ) λ 1 λ 2 /2 2 3 (1 ρ ) 2 1 + O(λ 4 ) . (11.5) K. Nakamura et al., JPG 37, 075021 (2010) (http://pdg.lbl.gov) July 30, 2010 14:36

Transcript of 11. THE CKM QUARK-MIXINGMATRIX11. CKM quark-mixing matrix 1 11. THE CKM QUARK-MIXINGMATRIX Revised...

Page 1: 11. THE CKM QUARK-MIXINGMATRIX11. CKM quark-mixing matrix 1 11. THE CKM QUARK-MIXINGMATRIX Revised February 2010 by A. Ceccucci (CERN), Z. Ligeti (LBNL), and Y. Sakai (KEK). 11.1.

11. CKM quark-mixing matrix 1

11. THE CKM QUARK-MIXING MATRIXRevised February 2010 by A. Ceccucci (CERN), Z. Ligeti (LBNL), and Y. Sakai (KEK).

11.1. Introduction

The masses and mixings of quarks have a common origin in the Standard Model (SM).They arise from the Yukawa interactions with the Higgs condensate,

LY = −Y dij QI

Li φ dIRj − Y u

ij QILi ε φ∗uI

Rj + h.c., (11.1)

where Y u,d are 3× 3 complex matrices, φ is the Higgs field, i, j are generation labels, andε is the 2 × 2 antisymmetric tensor. QI

L are left-handed quark doublets, and dIR and uI

Rare right-handed down- and up-type quark singlets, respectively, in the weak-eigenstatebasis. When φ acquires a vacuum expectation value, 〈φ〉 = (0, v/

√2), Eq. (11.1) yields

mass terms for the quarks. The physical states are obtained by diagonalizing Y u,d

by four unitary matrices, Vu,dL,R, as M

fdiag = V

fL Y f V

f†R (v/

√2), f = u, d. As a result,

the charged-current W± interactions couple to the physical uLj and dLk quarks withcouplings given by

VCKM ≡ V uL V

d†L =

⎛⎝ Vud Vus Vub

Vcd Vcs VcbVtd Vts Vtb

⎞⎠ . (11.2)

This Cabibbo-Kobayashi-Maskawa (CKM) matrix [1,2] is a 3 × 3 unitary matrix. Itcan be parameterized by three mixing angles and the CP -violating KM phase [2]. Ofthe many possible conventions, a standard choice has become [3]

V =

⎛⎝ c12c13 s12c13 s13e−iδ

−s12c23−c12s23s13eiδ c12c23−s12s23s13eiδ s23c13

s12s23−c12c23s13eiδ −c12s23−s12c23s13eiδ c23c13

⎞⎠ , (11.3)

where sij = sin θij , cij = cos θij , and δ is the phase responsible for all CP -violatingphenomena in flavor-changing processes in the SM. The angles θij can be chosen to lie inthe first quadrant, so sij , cij ≥ 0.

It is known experimentally that s13 � s23 � s12 � 1, and it is convenient to exhibitthis hierarchy using the Wolfenstein parameterization. We define [4–6]

s12 = λ =|Vus|√

|Vud|2 + |Vus|2, s23 = Aλ2 = λ

∣∣∣∣ Vcb

Vus

∣∣∣∣ ,

s13eiδ = V ∗

ub = Aλ3(ρ + iη) =Aλ3(ρ + iη)

√1 − A2λ4

√1 − λ2[1 − A2λ4(ρ + iη)]

. (11.4)

These relations ensure that ρ+ iη = −(VudV ∗ub)/(VcdV

∗cb) is phase-convention-independent,

and the CKM matrix written in terms of λ, A, ρ, and η is unitary to all orders in λ.The definitions of ρ, η reproduce all approximate results in the literature. For example,ρ = ρ(1 − λ2/2 + . . .) and we can write VCKM to O(λ4) either in terms of ρ, η or,traditionally,

V =

⎛⎝ 1 − λ2/2 λ Aλ3(ρ − iη)

−λ 1 − λ2/2 Aλ2

Aλ3(1 − ρ − iη) −Aλ2 1

⎞⎠ + O(λ4) . (11.5)

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2 11. CKM quark-mixing matrix

Figure 11.1: Sketch of the unitarity triangle.

The CKM matrix elements are fundamental parameters of the SM, so their precisedetermination is important. The unitarity of the CKM matrix imposes

∑i VijV

∗ik = δjk

and∑

j VijV∗kj = δik. The six vanishing combinations can be represented as triangles in

a complex plane, of which the ones obtained by taking scalar products of neighboringrows or columns are nearly degenerate. The areas of all triangles are the same, half ofthe Jarlskog invariant, J [7], which is a phase-convention-independent measure of CPviolation, defined by Im

[VijVklV

∗il V

∗kj

]= J

∑m,n εikmεjln.

The most commonly used unitarity triangle arises from

Vud V ∗ub + Vcd V ∗

cb + Vtd V ∗tb = 0 , (11.6)

by dividing each side by the best-known one, VcdV∗cb (see Fig. 1). Its vertices are exactly

(0, 0), (1, 0), and, due to the definition in Eq. (11.4), (ρ, η). An important goal offlavor physics is to overconstrain the CKM elements, and many measurements can beconveniently displayed and compared in the ρ, η plane.

Processes dominated by loop contributions in the SM are sensitive to new physics, andcan be used to extract CKM elements only if the SM is assumed. In Sec. 11.2 and 11.3,we describe such measurements assuming the SM, we give the global fit results for theCKM elements in Sec. 11.4, and discuss implications for new physics in Sec. 11.5.

11.2. Magnitudes of CKM elements

11.2.1. |Vud| :

The most precise determination of |Vud| comes from the study of superallowed 0+ → 0+

nuclear beta decays, which are pure vector transitions. Taking the average of the twentymost precise determinations [8] yields

|Vud| = 0.97425± 0.00022. (11.7)

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11. CKM quark-mixing matrix 3

The error is dominated by theoretical uncertainties stemming from nuclear Coulombdistortions and radiative corrections. A precise determination of |Vud| is also obtainedfrom the measurement of the neutron lifetime. The theoretical uncertainties are verysmall, but the determination is limited by the knowledge of the ratio of the axial-vector and vector couplings, gA = GA/GV [9]. The PIBETA experiment [10] hasimproved the measurement of the π+ → π0e+ν branching ratio to 0.6%, and quote|Vud| = 0.9728 ± 0.0030, in agreement with the more precise result listed above. Theinterest in this measurement is that the determination of |Vud| is very clean theoretically,because it is a pure vector transition and is free from nuclear-structure uncertainties.

11.2.2. |Vus| :The product of |Vus| and the form factor at q2 = 0, |Vus| f+(0) have been extracted

traditionally from K0L → πeν decays in order to avoid isospin-breaking corrections (π0−η

mixing) that affect K± semileptonic decay, and the complications induced by a second(scalar) form factor present in the muonic decays. The last round of experiments haslead to enough experimental constraints to justify the comparison between different decaymodes. Systematic errors related to the experimental quantities, e.g., the lifetime ofneutral or charged kaons, and the form factor determinations for electron and muonicdecays, differ among decay modes, and the consistency between different determinationsenhances the confidence in the final result. For this reason, we follow the prescription [11]to average K0

L → πeν, K0L → πμν, K± → π0e±ν, K± → π0μ±ν and K0

S → πeν.The average of these five decay modes yields |Vus| f+(0) = 0.21664 ± 0.00048. Resultsobtained from each decay mode, and exhaustive references to the experimental data, arelisted for instance in Ref. [9]. The form factor value f+(0) = 0.9644 ± 0.0049 [12] from athree-flavor unquenched lattice QCD calculation gives [9] |Vus| = 0.2246 ± 0.0012. Thebroadly used classic calculation of f+(0) [13] is in good agreement with this value, whileother calculations [14] differ by as much as 2%.

The calculation of the ratio of the kaon and pion decay constants enables one toextract |Vus/Vud| from K → μν(γ) and π → μν(γ), where (γ) indicates that radiativedecays are included [15]. The KLOE measurement of the K+ → μ+ν(γ) branchingratio [16], combined with the lattice QCD calculation, fK/fπ = 1.189 ± 0.007 [17], leadsto |Vus| = 0.2259 ± 0.0014, where the accuracy is limited by the knowledge of the ratio ofthe decay constants. The average of these two determinations is quoted by Ref. [9] as

|Vus| = 0.2252 ± 0.0009. (11.8)

The latest determination from hyperon decays can be found in Ref. [19]. The authorsfocus on the analysis of the vector form factor, protected from first order SU(3) breakingeffects by the Ademollo-Gatto theorem [20], and treat the ratio between the axial andvector form factors g1/f1 as experimental input, thus avoiding first order SU(3) breakingeffects in the axial-vector contribution. They find |Vus| = 0.2250 ± 0.0027, althoughthis does not include an estimate of the theoretical uncertainty due to second-orderSU(3) breaking, contrary to Eq. (11.8). Concerning hadronic τ decays to strangeparticles, the latest determinations based on LEP, and recent BABAR and Belle data yield|Vus| = 0.2208 ± 0.0039 [21]. A recent measurement of the ratio of branching fractions

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4 11. CKM quark-mixing matrix

B(τ → Kν)/B(τ → πν) by BABAR [22] combined with the above fK/fπ value gives|Vus| = 0.2255 ± 0.0024.

11.2.3. |Vcd| :The magnitude of Vcd can be extracted from semileptonic charm decays if theoretical

knowledge of the form factors is available. Three-flavor unquenched lattice QCDcalculations for D → K�ν and D → π�ν have been published [23]. Using these estimatesand the average of recent CLEO-c [24] and Belle [25] measurements of D → π�ν decays,one obtains |Vcd| = 0.229± 0.006± 0.024, where the first uncertainty is experimental, andthe second is from the theoretical uncertainty of the form factor.

This determination is not yet as precise as the one based on neutrino and antineutrinointeractions. The difference of the ratio of double-muon to single-muon production byneutrino and antineutrino beams is proportional to the charm cross section off valenced-quarks, and therefore to |Vcd|2 times the average semileptonic branching ratio of charmmesons, Bμ. The method was used first by CDHS [26] and then by CCFR [27,28] andCHARM II [29]. Averaging these results is complicated, not only because it requiresassumptions about the scale of the QCD corrections, but also because Bμ is an effectivequantity, which depends on the specific neutrino beam characteristics. Given that nonew experimental input is available, we quote the average provided in a previousreview, Bμ|Vcd|2 = (0.463 ± 0.034) × 10−2 [30]. Analysis cuts make these experimentsinsensitive to neutrino energies smaller than 30 GeV. Thus, Bμ should be computedusing only neutrino interactions with visible energy larger than 30GeV. An appraisal [31]based on charm-production fractions measured in neutrino interactions [32,33] givesBμ = 0.088 ± 0.006. Data from the CHORUS experiment [34] are sufficiently precise toextract Bμ directly, by comparing the number of charm decays with a muon to the totalnumber of charmed hadrons found in the nuclear emulsions. Requiring the visible energyto be larger than 30GeV, CHORUS finds Bμ = 0.085 ± 0.009 ± 0.006. To extract |Vcd|,we use the average of these two determinations, Bμ = 0.087 ± 0.005, and obtain

|Vcd| = 0.230 ± 0.011. (11.9)

11.2.4. |Vcs| :The determination of |Vcs| from neutrino and antineutrino scattering suffers from the

uncertainty of the s-quark sea content. Measurements sensitive to |Vcs| from on-shell W±decays were performed at LEP-2. The branching ratios of the W depend on the six CKMmatrix elements involving quarks with masses smaller than MW . The W branching ratioto each lepton flavor is given by 1/B(W → �ν�) = 3

[1 +

∑u,c,d,s,b |Vij |2 (1 + αs(mW )/π)

].

The measurement assuming lepton universality, B(W → �ν�) = (10.83±0.07±0.07) % [35],implies

∑u,c,d,s,b |Vij |2 = 2.002 ± 0.027. This is a precise test of unitarity, but only

flavor-tagged W -decay measurements determine |Vcs| directly. DELPHI measured taggedW+ → cs decays, obtaining |Vcs| = 0.94+0.32

−0.26 ± 0.13 [36]. Hereafter, the first error isstatistical and the second is systematic, unless mentioned otherwise.

The direct determination of |Vcs| is possible from semileptonic D or leptonic Ds

decays, using unquenched lattice QCD calculations of the semileptonic D form factor or

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11. CKM quark-mixing matrix 5

the Ds decay constant. For muonic decays, the average of Belle [37] and CLEO-c [38]gives B(D+

s → μ+ν) = (5.81 ± 0.43) × 10−3 [39]. The ajusted BABAR [40] measurementgives B(D+

s → μ+ν) = (4.81 ± 0.68) × 10−3 [39]. For decays with τ leptons, the averageof recent CLEO-c measurements [38,41] gives B(D+

s → τ+ν) = (5.61 ± 0.44) × 10−2 [39].From each of these values, determinations of |Vcs| can be obtained by using thePDG values for the mass and lifetime of the Ds, the masses of the leptons, andfDs = (242.8 ± 3.2 ± 5.3)MeV [42]. The average of these three determinations gives|Vcs| = 1.030 ± 0.038, where the error is dominated by the lattice QCD determination offDs . In semileptonic D decays, unquenched lattice QCD calculations have predicted thenormalization and the shape (dependence on the invariant mass of the lepton pair, q2)of the form factors in D → K�ν and D → π�ν [23]. Using these theoretical results andthe average of recent CLEO-c [24], Belle [25] and BABAR [43] measurements of B → K�νdecays, one obtains |Vcs| = 0.98 ± 0.01 ± 0.10, where the first error is experimental andthe second, which is dominant, is from the theoretical uncertainty of the form factor.Averaging the determinations from leptonic and semileptonic decays, we find

|Vcs| = 1.023 ± 0.036. (11.10)

11.2.5. |Vcb| :This matrix element can be determined from exclusive and inclusive semileptonic decays

of B mesons to charm. The inclusive determinations use the semileptonic decay ratemeasurement, together with the leptonic energy and the hadronic invariant-mass spectra.The theoretical foundation of the calculation is the operator product expansion [44,45]. Itexpresses the total rate and moments of differential energy and invariant-mass spectra asexpansions in αs, and inverse powers of the heavy quark mass. The dependence on mb,mc, and the parameters that occur at subleading order is different for different moments,and a large number of measured moments overconstrains all the parameters, and teststhe consistency of the determination. The precise extraction of |Vcb| requires using a“threshold” quark mass definition [46,47]. Inclusive measurements have been performedusing B mesons from Z0 decays at LEP, and at e+e− machines operated at the Υ(4S). AtLEP, the large boost of B mesons from the Z0 allows the determination of the momentsthroughout phase space, which is not possible otherwise, but the large statistics availableat the B factories lead to more precise determinations. An average of the measurementsand a compilation of the references are provided by Ref. [48]: |Vcb| = (41.5± 0.7)× 10−3.

Exclusive determinations are based on semileptonic B decays to D and D∗. In themb,c ΛQCD limit, all form factors are given by a single Isgur-Wise function [49], whichdepends on the product of the four-velocities of the B and D(∗) mesons, w = v · v′.Heavy quark symmetry determines the normalization of the rate at w = 1, the maximummomentum transfer to the leptons, and |Vcb| is obtained from an extrapolation tow = 1. The exclusive determination, |Vcb| = (38.7 ± 1.1) × 10−3 [48], is less precisethan the inclusive one because of the theoretical uncertainty in the form factor and theexperimental uncertainty in the rate near w = 1. Ref. [48] quotes a combination with ascaled error as

|Vcb| = (40.6 ± 1.3) × 10−3. (11.11)

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6 11. CKM quark-mixing matrix

11.2.6. |Vub| :The determination of |Vub| from inclusive B → Xu�ν decay suffers from large B → Xc�ν

backgrounds. In most regions of phase space where the charm background is kinematicallyforbidden, the hadronic physics enters via unknown nonperturbative functions, so-calledshape functions. (In contrast, the nonperturbative physics for |Vcb| is encoded in a fewparameters.) At leading order in ΛQCD/mb, there is only one shape function, whichcan be extracted from the photon energy spectrum in B → Xsγ [50,51], and applied toseveral spectra in B → Xu�ν. The subleading shape functions are modeled in the currentdeterminations. Phase space cuts for which the rate has only subleading dependence onthe shape function are also possible [52]. The measurements of both the hadronic and theleptonic systems are important for an optimal choice of phase space. A different approachis to make the measurements more inclusive by extending them deeper into the B → Xc�νregion, and thus reduce the theoretical uncertainties. Analyses of the electron-energyendpoint from CLEO [53], BABAR [54], and Belle [55] quote B → Xueν partial rates for|�pe| ≥ 2.0GeV and 1.9GeV, which are well below the charm endpoint. The large andpure BB samples at the B factories permit the selection of B → Xu�ν decays in eventswhere the other B is fully reconstructed [56]. With this full-reconstruction tag method,the four-momenta of both the leptonic and the hadronic systems can be measured. Italso gives access to a wider kinematic region because of improved signal purity. Ref. [48]quotes an inclusive average as |Vub| = (4.27 ± 0.38) × 10−3.

To extract |Vub| from an exclusive channel, the form factors have to be known.Experimentally, better signal-to-background ratios are offset by smaller yields. TheB → π�ν branching ratio is now known to 5%. Unquenched lattice QCD calculations ofthe B → π�ν form factor are available [57,58] for the high q2 region (q2 > 16 or 18 GeV2).A simultaneous fit to the experimental partial rates and lattice points versus q2

yields |Vub| = (3.38 ± 0.36) × 10−3 [58]. Light-cone QCD sum rules are applicable forq2 < 14 GeV2 [59] and yield similar results.

The theoretical uncertainties in extracting |Vub| from inclusive and exclusive decaysare different. A combination of the determinations is quoted by Ref. [48] as

|Vub| = (3.89 ± 0.44) × 10−3. (11.12)

11.2.7. |Vtd| and |Vts| :The CKM elements |Vtd| and |Vts| cannot be measured from tree-level decays of

the top quark, so one has to rely on determinations from B–B oscillations mediatedby box diagrams with top quarks, or loop-mediated rare K and B decays. Theoreticaluncertainties in hadronic effects limit the accuracy of the current determinations. Thesecan be reduced by taking ratios of processes that are equal in the flavor SU(3) limit todetermine |Vtd/Vts|.

The mass difference of the two neutral B meson mass eigenstates is very wellmeasured, Δmd = (0.507 ± 0.005) ps−1 [60]. For the B0

s system, CDF measuredΔms = (17.77 ± 0.10 ± 0.07) ps−1 [61] with more than 5σ significance (the DØ result [62]is compatible and has about 2σ significance). Using the unquenched lattice QCD

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11. CKM quark-mixing matrix 7

calculations [63] fBd

√BBd

= (216 ± 9 ± 13)MeV, fBs

√BBs = (275 ± 7 ± 13)MeV, and

assuming |Vtb| = 1, one finds

|Vtd| = (8.4 ± 0.6) × 10−3, |Vts| = (38.7 ± 2.1) × 10−3. (11.13)

The uncertainties are dominated by lattice QCD. Several uncertainties are reduced in

the calculation of the ratio ξ =(fBs

√BBs

)/(fBd

√BBd

)= 1.243 ± 0.021 ± 0.021, and

therefore the constraint on |Vtd/Vts| from Δmd/Δms is more reliable theoretically. Theseprovide a new, theoretically clean, and significantly improved constraint∣∣Vtd/Vts

∣∣ = 0.211 ± 0.001 ± 0.005. (11.14)

The inclusive branching ratio B(B → Xsγ) = (3.52 ± 0.25) × 10−4 extrapolated toEγ > E0 = 1.6 GeV [64] is also sensitive to VtbV

∗ts. In addition to t-quark penguins, a

large part of the sensitivity comes from charm contributions proportional to VcbV∗cs via

the application of 3 × 3 CKM unitarity (which is used here; any CKM determinationfrom loop processes necessarily assumes the SM). With the NNLO calculation ofB(B → Xsγ)Eγ>E0

/B(B → Xceν) [65], we obtain |Vts/Vcb| = (1.04 ± 0.05).A complementary determination of |Vtd/Vts| is possible from the ratio of B → ργ

and K∗γ rates. The ratio of the neutral modes is theoretically cleaner than that of thecharged ones, because the poorly known spectator-interaction contribution is expectedto be smaller (W -exchange vs. weak annihilation). For now, because of low statistics weaverage the charged and neutral rates assuming the isospin symmetry and heavy quarklimit motivated relation, |Vtd/Vts|2/ξ2

γ = [Γ(B+ → ρ+γ) + 2Γ(B0 → ρ0γ)]/[Γ(B+ →K∗+γ) + Γ(B0 → K∗0γ)] = (3.19 ± 0.46)% [64]. Here ξγ contains the poorly knownhadronic physics. Using ξγ = 1.2 ± 0.2 [66], and combining the experimental andtheoretical errors in quadrature, gives |Vtd/Vts| = 0.21 ± 0.04.

A theoretically clean determination of |VtdV∗ts| is possible from K+ → π+νν decay [67].

Experimentally, only seven events have been observed [68] and the rate is consistent withthe SM with large uncertainties. Much more data are needed for a precision measurement.

11.2.8. |Vtb| :The determination of |Vtb| from top decays uses the ratio of branching fractions

R = B(t → Wb)/B(t → Wq) = |Vtb|2/(∑

q |Vtq|2) = |Vtb|2, where q = b, s, d. The CDFand DØ measurements performed on data collected during Run II of the Tevatron give|Vtb| > 0.78 [69] and |Vtb| > 0.89 [70], respectively, at 95% CL. The direct determinationof |Vtb| without assuming unitarity is possible from the single top-quark-production crosssection. The (2.76+0.58

−0.47) pb [71] average cross section measured by DØ [72] and CDF [73]implies

|Vtb| = 0.88 ± 0.07 . (11.15)

An attempt at constraining |Vtb| from the precision electroweak data was madein [74]. The result, mostly driven by the top-loop contributions to Γ(Z → bb), gives|Vtb| = 0.77+0.18

−0.24.

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8 11. CKM quark-mixing matrix

11.3. Phases of CKM elements

As can be seen from Fig. 11.1, the angles of the unitarity triangle are

β = φ1 = arg(−VcdV

∗cb

VtdV∗tb

),

α = φ2 = arg(− VtdV

∗tb

VudV ∗ub

),

γ = φ3 = arg(−VudV ∗

ub

VcdV ∗cb

). (11.16)

Since CP violation involves phases of CKM elements, many measurements of CP -violatingobservables can be used to constrain these angles and the ρ, η parameters.

11.3.1. ε and ε′ :The measurement of CP violation in K0–K0 mixing, |ε| = (2.233± 0.015)× 10−3 [75],

provides important information about the CKM matrix. In the SM, in the basis whereVudV ∗

us is real [76]

|ε| =G2

F f2KmKm2

W

12√

2 π2ΔmK

BK

{η1S(xc) Im[(VcsV

∗cd)2]

+ η2S(xt) Im[(VtsV∗td)2] + 2η3S(xc, xt) Im(VcsV

∗cdVtsV

∗td)

}, (11.17)

where S is an Inami-Lim function [77], xq = m2q/m2

W , and ηi are perturbativeQCD corrections. The constraint from ε in the ρ, η plane is bounded by approximatehyperbolas. The dominant uncertainties are due to the bag parameter, for which we useBK = 0.725 ± 0.026 from lattice QCD [63], and the parametric uncertainty proportionalto σ(A4) from (VtsV

∗td)2, which is approximately σ(|Vcb|4).

The measurement of 6 Re(ε′/ε) = 1 − |η00/η+−|2, where η00 and η+− are theCP -violating amplitude ratios of K0

S and K0L decays to two pions, provides a

qualitative test of the CKM mechanism. Its nonzero experimental average, Re(ε′/ε) =(1.67 ± 0.23) × 10−3 [75], demonstrates the existence of direct CP violation, a predictionof the KM ansatz. While Re(ε′/ε) ∝ Im(VtdV

∗ts), this quantity cannot easily be used to

extract CKM parameters, because the electromagnetic penguin contributions tend tocancel the gluonic penguins for large mt [78], thereby significantly increasing the hadronicuncertainties. Most estimates [79–82] agree with the observed value, indicating that η ispositive. Progress in lattice QCD, in particular finite-volume calculations [83,84], mayeventually provide a determination of the K → ππ matrix elements.

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11. CKM quark-mixing matrix 9

11.3.2. β / φ1 :

11.3.2.1. Charmonium modes:CP -violation measurements in B-meson decays provide direct information on the

angles of the unitarity triangle, shown in Fig. 11.1. These overconstraining measurementsserve to improve the determination of the CKM elements, or to reveal effects beyondthe SM.

The time-dependent CP asymmetry of neutral B-decays to a final state f common toB0 and B0 is given by [85,86]

Af =Γ(B0(t) → f) − Γ(B0(t) → f)Γ(B0(t) → f) + Γ(B0(t) → f)

= Sf sin(Δmd t) − Cf cos(Δmd t), (11.18)

where

Sf =2 Imλf

1 + |λf |2, Cf =

1 − |λf |21 + |λf |2

, λf =q

p

Af

Af. (11.19)

Here, q/p describes B0–B0 mixing and, to a good approximation in the SM, q/p =V ∗

tbVtd/VtbV∗td = e−2iβ+O(λ4) in the usual phase convention. Af (Af ) is the amplitude of

the B0 → f (B0 → f) decay. If f is a CP eigenstate, and amplitudes with one CKMphase dominate the decay, then |Af | = |Af |, Cf = 0, and Sf = sin(arg λf ) = ηf sin 2φ,where ηf is the CP eigenvalue of f and 2φ is the phase difference between the B0 → f

and B0 → B0 → f decay paths. A contribution of another amplitude to the decay with adifferent CKM phase makes the value of Sf sensitive to relative strong interaction phasesbetween the decay amplitudes (it also makes Cf �= 0 possible).

The b → ccs decays to CP eigenstates (B0 → charmonium K0S,L) are the theoretically

cleanest examples, measuring Sf = −ηf sin 2β. The b → sqq penguin amplitudes havedominantly the same weak phase as the b → ccs tree amplitude. Since only λ2-suppressedpenguin amplitudes introduce a new CP -violating phase, amplitudes with a single weakphase dominate, and we expect

∣∣|AψK/AψK | − 1∣∣ < 0.01. The e+e− asymmetric-energy

B-factory experiments, BABAR [88] and Belle [89], provide precise measurements. Theworld average is [64]

sin 2β = 0.673 ± 0.023 . (11.20)

This measurement has a four-fold ambiguity in β, which can be resolved by a globalfit as mentioned in Sec. 11.4. Experimentally, the two-fold ambiguity β → π/2 − β(but not β → π + β) can be resolved by a time-dependent angular analysis ofB0 → J/ψK∗0 [90,91], or a time-dependent Dalitz plot analysis of B0 → D0h0

(h0 = π0, η, ω) with D0 → K0Sπ+π− [92,93]. These results indicate that negative cos 2β

solutions are very unlikely, in agreement with the global CKM fit result.The b → ccd mediated transitions, such as B0 → J/ψπ0 and B0 → D(∗)+D(∗)−,

also measure approximately sin 2β. However, the dominant component of the b → dpenguin amplitude has a different CKM phase (V ∗

tbVtd) than the tree amplitude (V ∗cbVcd),

and its magnitudes are of the same order in λ. Therefore, the effect of penguins could

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10 11. CKM quark-mixing matrix

be large, resulting in Sf �= −ηf sin 2β and Cf �= 0. These decay modes have alsobeen measured by BABAR and Belle. The world averages [64], SJ/ψπ0 = −0.93 ± 0.15,SD+D− = −0.89 ± 0.26, and SD∗+D∗− = −0.77 ± 0.14 (ηf = +1 for these modes), areconsistent with sin 2β obtained from B0 → charmonium K0 decays, and the Cf ’s areconsistent with zero, although the uncertainties are sizable.

The b → cud decays, B0 → D0h0 with D0 → CP eigenstates, have no penguincontributions and provide theoretically clean sin 2β measurements. BABAR measuredS

D(∗)h0 = −0.56 ± 0.25 [87].

11.3.2.2. Penguin-dominated modes:The b → sqq penguin-dominated decays have the same CKM phase as the b → ccs

tree level decays, up to corrections suppressed by λ2, since V ∗tbVts = −V ∗

cbVcs[1 + O(λ2)].Therefore, decays such as B0 → φK0 and η′K0 provide sin 2β measurements in the SM.Any new physics contribution to the amplitude with a different weak phase would giverise to Sf �= −ηf sin 2β, and possibly Cf �= 0. Therefore, the main interest in these modesis not simply to measure sin 2β, but to search for new physics. Measurements of manyother decay modes in this category, such as B → π0K0

S , K0SK0

SK0S , etc., have also been

performed by BABAR and Belle. The results and their uncertainties are summarized inFig. 12.3 and Table 12.1 of Ref. [86].

11.3.3. α / φ2 :Since α is the phase between V ∗

tbVtd and V ∗ubVud, only time-dependent CP asymmetries

in b → uud decay dominated modes can directly measure sin 2α, in contrast to sin 2β,where several different transitions can be used. Since b → d penguin amplitudes have adifferent CKM phase than b → uud tree amplitudes, and their magnitudes are of thesame order in λ, the penguin contribution can be sizable, which makes the determinationof α complicated. To date, α has been measured in B → ππ, ρπ and ρρ decay modes.

11.3.3.1. B → ππ:It is now experimentally well established that there is a sizable contribution of b → d

penguin amplitudes in B → ππ decays. Thus, Sπ+π− in the time-dependent B0 → π+π−analysis does not measure sin 2α, but

Sπ+π− =√

1 − C2π+π− sin(2α + 2Δα), (11.21)

where 2Δα is the phase difference between e2iγAπ+π− and Aπ+π− . The value of Δα,hence α, can be extracted using the isospin relation among the amplitudes of B0 → π+π−,B0 → π0π0, and B+ → π+π0 decays [94],

1√2

Aπ+π− + Aπ0π0 − Aπ+π0 = 0, (11.22)

and a similar expression for the Aππ’s. This method utilizes the fact that a pair ofpions from B → ππ decay must be in a zero angular momentum state, and, because of

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11. CKM quark-mixing matrix 11

Bose statistics, they must have even isospin. Consequently, π0π± is in a pure isospin-2state, while the penguin amplitudes only contribute to the isospin-0 final state. Thelatter does not hold for the electroweak penguin amplitudes, but their effect is expectedto be small. The isospin analysis uses the world averages [64] Sπ+π− = −0.65 ± 0.07,Cπ+π− = −0.38 ± 0.06, the branching fractions of all three modes, and the directCP asymmetry Cπ0π0 = −0.43+0.25

−0.24. This analysis leads to 16 mirror solutions for0 ≤ α < 2π. Because of this, and the sizable experimental error of the B0 → π0π0

rate and CP asymmetry, only a loose constraint on α can be obtained at present [95],0◦ < α < 7◦, 81◦ < α < 103◦, 121◦ < α < 150◦, and 166◦ < α < 180◦ at 68% CL.

11.3.3.2. B → ρρ:The decay B0 → ρ+ρ− contains two vector mesons in the final state, which in general

is a mixture of CP -even and CP -odd components. Therefore, it was thought thatextracting α from this mode would be complicated.

However, the longitudinal polarization fractions (fL) in B+ → ρ+ρ0 and B0 → ρ+ρ−decays were measured to be close to unity [96], which implies that the final states arealmost purely CP -even. Furthermore, B(B0 → ρ0ρ0) = (0.73+0.27

−0.28)×10−6 is much smallerthan B(B0 → ρ+ρ−) = (24.2+3.1

−3.2) × 10−6 and B(B+ → ρ+ρ0) = (24.0+1.9−2.0) × 10−6 [64],

which implies that the effect of the penguin diagrams is small. The isospin analysis usingthe world averages, Sρ+ρ− = −0.05 ± 0.17 and Cρ+ρ− = −0.06 ± 0.13 [64], together withthe time-dependent CP asymmetry, Sρ0ρ0 = −0.3 ± 0.7 and Cρ0ρ0 = −0.2 ± 0.9 [97],and the above-mentioned branching fractions, gives α = (89.9 ± 5.4)◦ [95], with a mirrorsolution at 3π/2 − α. A possible small violation of Eq. (11.22) due to the finite width ofthe ρ [98] is neglected.

11.3.3.3. B → ρπ:The final state in B0 → ρ+π− decay is not a CP eigenstate, but this decay proceeds

via the same quark-level diagrams as B0 → π+π−, and both B0 and B0 can decay toρ+π−. Consequently, mixing-induced CP violations can occur in four decay amplitudes,B0 → ρ±π∓ and B0 → ρ±π∓. The time-dependent Dalitz plot analysis of B0 → π+π−π0

decays permits the extraction of α with a single discrete ambiguity, α → α + π, sinceone knows the variation of the strong phases in the interference regions of the ρ+π−,ρ−π+, and ρ0π0 amplitudes in the Dalitz plot [99]. The combination of Belle [100] andBABAR [101] measurements gives α = (120+11

−7 )◦ [95]. This constraint is still moderate,and there are also solutions around 30◦ and 90◦ within 2σ significance level.

Combining the above-mentioned three decay modes [95], α is constrained as

α = (89.0+4.4−4.2)

◦. (11.23)

A different statistical approach [102] gives similar constraint from the combination ofthese measurements.

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12 11. CKM quark-mixing matrix

11.3.4. γ / φ3 :By virtue of Eq. (11.16), γ does not depend on CKM elements involving the top quark,

so it can be measured in tree-level B decays. This is an important distinction from themeasurements of α and β, and implies that the measurements of γ are unlikely to beaffected by physics beyond the SM.

11.3.4.1. B± → DK±:The interference of B− → D0K− (b → cus) and B− → D0K− (b → ucs) transitions

can be studied in final states accessible in both D0 and D0 decays [85]. In principle, itis possible to extract the B and D decay amplitudes, the relative strong phases, and theweak phase γ from the data.

A practical complication is that the precision depends sensitively on the ratio of theinterfering amplitudes

rB =∣∣∣A(B− → D0K−)

/A(B− → D0K−)

∣∣∣ , (11.24)

which is around 0.1 − 0.2. The original GLW method [103,104] considers D decays toCP eigenstates, such as B± → D

(∗)CP (→ π+π−)K±(∗). To alleviate the smallness of

rB and make the interfering amplitudes (which are products of the B and D decayamplitudes) comparable in magnitude, the ADS method [105] considers final states whereCabibbo-allowed D0 and doubly-Cabibbo-suppressed D0 decays interfere. Extensivemeasurements have been made by the B factories using both methods [106].

It was realized that both D0 and D0 have large branching fractions to certainthree-body final states, such as KSπ+π−, and the analysis can be optimized by studyingthe Dalitz plot dependence of the interferences [107,108]. The best present determinationof γ comes from this method. Belle [109] and BABAR [110] obtained γ = (76+12

−13 ± 4 ± 9)◦and γ = (76± 22± 5± 5)◦, respectively, where the last uncertainty is due to the D-decaymodeling. The error is sensitive to the central value of the amplitude ratio rB (and r∗Bfor the D∗K mode), for which Belle found somewhat larger central values than BABAR.The same values of r

(∗)B enter the ADS analyses, and the data can be combined to fit for

r(∗)B and γ. The D0–D0-mixing has been neglected in all measurements, but its effect on

γ is far below the present experimental accuracy [111], unless D0–D0-mixing is due toCP -violating new physics, in which case it can be included in the analysis [112].

Combining the GLW, ADS, and Dalitz analyses [95], γ is constrained as

γ = (73+22−25)

◦. (11.25)

Similar results are found in [102].

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11. CKM quark-mixing matrix 13

11.3.4.2. B0 → D(∗)±π∓:

The interference of b → u and b → c transitions can be studied in B0 → D(∗)+π−(b → cud) and B0 → B0 → D(∗)+π− (b → ucd) decays and their CP conjugates, sinceboth B0 and B0 decay to D(∗)±π∓ (or D±ρ∓, etc.). Since there are only tree and nopenguin contributions to these decays, in principle, it is possible to extract from thefour time-dependent rates the magnitudes of the two hadronic amplitudes, their relativestrong phase, and the weak phase between the two-decay paths, which is 2β + γ.

A complication is that the ratio of the interfering amplitudes is very small,rDπ = A(B0 → D+π−)/A(B0 → D+π−) = O(0.01) (and similarly for rD∗π and rDρ),and therefore it has not been possible to measure it. To obtain 2β + γ, SU(3) flavorsymmetry and dynamical assumptions have been used to relate A(B0 → D−π+) toA(B0 → D−

s π+), so this measurement is not model-independent at present. Combiningthe D±π∓ D∗±π∓ and D±ρ∓ measurements [113] gives sin(2β+γ) > 0.68 at 68% CL [95],consistent with the previously discussed results for β and γ. The amplitude ratio is muchlarger in the analogous B0

s → D±s K∓ decays, so it will be possible at LHCb to measure

it and model-independently extract γ − 2βs [114] (where βs = arg(−VtsV∗tb/VcsV

∗cb) is

related to the phase of Bs mixing).

11.4. Global fit in the Standard Model

Using the independently measured CKM elements mentioned in the previous sections,the unitarity of the CKM matrix can be checked. We obtain |Vud|2 + |Vus|2 + |Vub|2 =0.9999 ± 0.0006 (1st row), |Vcd|2 + |Vcs|2 + |Vcb|2 = 1.101 ± 0.074 (2nd row), |Vud|2 +|Vcd|2+|Vtd|2 = 1.002±0.005 (1st column), and |Vus|2+|Vcs|2+|Vts|2 = 1.098±0.074 (2ndcolumn), respectively. The uncertainties in the second row and column are dominatedby that of |Vcs|. For the second row, a more stringent check is obtained from themeasurement of

∑u,c,d,s,b |Vij |2 in Sec. 11.2.4 minus the sum in the first row above:

|Vcd|2 + |Vcs|2 + |Vcb|2 = 1.002 ± 0.027. These provide strong tests of the unitarity of theCKM matrix. The sum of the three angles of the unitarity triangle, α+β+γ = (183+22

−25)◦,

is also consistent with the SM expectation.

The CKM matrix elements can be most precisely determined by a global fit thatuses all available measurements and imposes the SM constraints (i.e., three generationunitarity). The fit must also use theory predictions for hadronic matrix elements, whichsometimes have significant uncertainties. There are several approaches to combining theexperimental data. CKMfitter [6,95] and Ref. [115] (which develops [116,117] further) usefrequentist statistics, while UTfit [102,118] uses a Bayesian approach. These approachesprovide similar results.

The constraints implied by the unitarity of the three generation CKM matrixsignificantly reduce the allowed range of some of the CKM elements. The fit for theWolfenstein parameters defined in Eq. (11.4) gives

λ = 0.2253± 0.0007 , A = 0.808+0.022−0.015 ,

ρ = 0.132+0.022−0.014 , η = 0.341 ± 0.013 . (11.26)

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14 11. CKM quark-mixing matrix

γ

γ

α

α

dmΔKε

smΔ & dmΔ

ubV

βsin 2

(excl. at CL > 0.95) < 0βsol. w/ cos 2

excluded at CL > 0.95

α

βγ

ρ-1.0 -0.5 0.0 0.5 1.0 1.5 2.0

η

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5excluded area has CL > 0.95

Figure 11.2: Constraints on the ρ, η plane. The shaded areas have 95% CL. Colorversion at end of book.

These values are obtained using the method of Refs. [6,95]. Using the prescriptionof Refs. [102,118] gives λ = 0.2246 ± 0.0011, A = 0.832 ± 0.017, ρ = 0.130 ± 0.018,η = 0.350± 0.013 [119]. The fit results for the magnitudes of all nine CKM elements are.

VCKM =

⎛⎝ 0.97428± 0.00015 0.2253 ± 0.0007 0.00347+0.00016

−0.00012

0.2252 ± 0.0007 0.97345+0.00015−0.00016 0.0410+0.0011

−0.0007

0.00862+0.00026−0.00020 0.0403+0.0011

−0.0007 0.999152+0.000030−0.000045

⎞⎠ , (11.27)

and the Jarlskog invariant is J = (2.91+0.19−0.11) × 10−5.

Fig. 11.2 illustrates the constraints on the ρ, η plane from various measurements andthe global fit result. The shaded 95% CL regions all overlap consistently around theglobal fit region, though the consistency of |Vub/Vcb| and sin 2β is not very good.

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11. CKM quark-mixing matrix 15

11.5. Implications beyond the SM

The effects in B, K, and D decays and mixings due to high-scale physics (W , Z, t, hin the SM, and new physics particles) can be parameterized by operators made of SMfields, obeying the SU(3)× SU(2)× U(1) gauge symmetry. The non-SM contributions tothe coefficients of these operators are suppressed by powers of the scale of new physics.At lowest order, there are of order a hundred flavor-changing operators of dimension-6,and the observable effects of non-SM interactions are encoded in their coefficients. Inthe SM, these coefficients are determined by just the four CKM parameters, and theW , Z, and quark masses. For example, Δmd, Γ(B → ργ), and Γ(B → Xd�

+�−) are allproportional to |VtdV

∗tb|2 in the SM, however, they may receive unrelated contributions

from new physics. The new physics contributions may or may not obey the SM relations.(For example, the flavor sector of the MSSM contains 69 CP -conserving parameters and41 CP -violating phases, i.e., 40 new ones [120]). Thus, similar to the measurements ofsin 2β in tree- and loop-dominated decay modes, overconstraining measurements of themagnitudes and phases of flavor-changing neutral current amplitudes give good sensitivityto new physics.

To illustrate the level of suppression required for non-SM contributions, considera class of models in which the unitarity of the CKM matrix is maintained, and thedominant effect of new physics is to modify the neutral meson mixing amplitudes [121]by (zij/Λ2)(qiγ

μPLqj)2 (for recent reviews, see [122,123]). It is only known since themeasurements of γ and α that the SM gives the leading contribution to B0 –B0

mixing [6,124]. Nevertheless, new physics with a generic weak phase may still contributeto neutral meson mixings at a significant fraction of the SM [125,118]. The existingdata imply that Λ/|zij |1/2 has to exceed about 104 TeV for K0 –K0 mixing, 103 TeV forD0 –D0 mixing, 500TeV for B0 –B0 mixing, and 100TeV for B0

s –B0s mixing [118,123].

(Some other operators are even better constrained [118].) The constraints are thestrongest in the kaon sector, because the CKM suppression is the most severe. Thus, ifthere is new physics at the TeV scale, |zij | � 1 is required. Even if |zij | are suppressedby a loop factor and |V ∗

tiVtj |2 (in the down quark sector), similar to the SM, one expectspercent-level effects, which may be observable in forthcoming flavor physics experiments.To constrain such extensions of the SM, many measurements irrelevant for the SM-CKMfit, such as the CP asymmetry in semileptonic B decays [126], are important.

Many key measurements, which are sensitive to non-SM flavor physics, are not usefulto think about in terms of constraining CKM parameters. For example, besides theangles in Eq. (11.16), a key quantity in the Bs sector is βs = arg(−VtsV

∗tb/VcsV

∗cb), which

is the small, λ2-suppressed, angle of a “squashed” unitarity triangle, obtained by takingthe scalar product of the second and third columns. The angle βs can be measured viatime-dependent CP violation in B0

s → J/ψφ, similar to β in B0 → J/ψK0. Checking ifβs agrees with its SM prediction, βs = 0.018 ± 0.001 [95], is an equally important testof the theory. The first flavor-tagged time-dependent CP -asymmetry measurements ofB0

s → J/ψφ decay appeared recently [127], giving a mild hint of a possible deviation.In the kaon sector, the two measured CP -violating observables ε and ε′ are tiny,

so models in which all sources of CP violation are small were viable before the

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16 11. CKM quark-mixing matrix

B-factory measurements. Since the measurement of sin 2β, we know that CP violationcan be an O(1) effect, and only flavor mixing is suppressed between the three quarkgenerations. Thus, many models with spontaneous CP violation are excluded. In thekaon sector, a very clean test of the SM will come from measurements of K+ → π+ννand K0

L → π0νν. These loop-induced rare decays are sensitive to new physics, and willallow a determination of β independent of its value measured in B decays [128].

The CKM elements are fundamental parameters, so they should be measured asprecisely as possible. The overconstraining measurements of CP asymmetries, mixing,semileptonic, and rare decays have started to severely constrain the magnitudes andphases of possible new physics contributions to flavor-changing interactions. When newparticles are observed at the LHC, it will be important to know the flavor parameters asprecisely as possible to understand the underlying physics.

References:1. N. Cabibbo, Phys. Rev. Lett. 10, 531 (1963).2. M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49, 652 (1973).3. L. L. Chau and W. Y. Keung, Phys. Rev. Lett. 53, 1802 (1984).4. L. Wolfenstein, Phys. Rev. Lett. 51, 1945 (1983).5. A. J. Buras et al., Phys. Rev. D50, 3433 (1994) [hep-ph/9403384].6. J. Charles et al. [CKMfitter Group], Eur. Phys. J. C41, 1 (2005) [hep-ph/0406184].7. C. Jarlskog, Phys. Rev. Lett. 55, 1039 (1985).8. J. C. Hardy and I. S. Towner, Phys. Rev. C70, 055502 (2009) [arXiv:0812.1202

[nucl-ex]].9. E. Blucher and W.J. Marciano, “Vud, Vus, the Cabibbo Angle and CKM Unitarity,”

in this Review.10. D. Pocanic et al., Phys. Rev. Lett. 93, 181803 (2004) [hep-ex/0312030].11. M. Antonelli et al. [The FlaviaNet Kaon Working Group], arXiv:0801.1817; see

also http://www.lnf.infn.it/wg/vus.12. P. A. Boyle et al., Phys. Rev. Lett. 100, 141601 (2008) [arXiv:0710.5136].13. H. Leutwyler and M. Roos, Z. Phys. C25, 91 (1984).14. J. Bijnens and P. Talavera, Nucl. Phys. B669, 341 (2003) [hep-ph/0303103];

M. Jamin et al., JHEP 402, 047 (2004) [hep-ph/0401080];V. Cirigliano et al., JHEP 504, 6 (2005) [hep-ph/0503108];C. Dawson et al., PoS LAT2005, 337 (2005) [hep-lat/0510018];N. Tsutsui et al. [JLQCD Collab.], PoS LAT2005, 357 (2005) [hep-lat/0510068];M. Okamoto [Fermilab Lattice Collab.], hep-lat/0412044.

15. W. J. Marciano, Phys. Rev. Lett. 93, 231803 (2004) [hep-ph/0402299].16. F. Ambrosino et al. [KLOE Collab.], Phys. Lett. B632, 76 (2006) [hep-ex/0509045].17. E. Follana et al. [HPQCD and UKQCD Collab.s], Phys. Rev. Lett. 100, 062002

(2008) [arXiv:0706.1726].18. C. Bernard et al. [MILC Collab.], PoS LAT2007, 090 (2006)[arXiv:0710.1118].19. N. Cabibbo et al., Ann. Rev. Nucl. and Part. Sci. 53, 39 (2003) [hep-ph/0307298];

Phys. Rev. Lett. 92, 251803 (2004) [hep-ph/0307214].20. M. Ademollo and R. Gatto, Phys. Rev. Lett. 13, 264 (1964).

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11. CKM quark-mixing matrix 17

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